On curves over finite fields with many rational points

We study arithmetical and geometrical properties of {\it maximal curves}, that is, curves defined over the finite field $\mathbb F_{q^2}$ whose number of $\mathbb F_{q^2}$-rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are $\mathbb F_{q^2}$-isomorphic to $y^q+y=x^m$ for some $m\in \mathbb Z^+$.


Introduction
Goppa in [Go] showed how to construct linear codes from curves defined over finite fields.One of the main features of these codes is the fact that one can state a lower bound for the minimum distance of the codes.In fact, let C X (D, G) be a Goppa code defined over a curve X over the finite field F q with q elements, where D = P 1 + . . .+ P n , P i ∈ X(F q ) for each i and G is a F q -rational divisor on X.Then it is known that the minimum distance d of C X (D, G) satisfies Certainly this bound is meaningful only if n is large enough.This provides motivation for the study of curves over finite fields with many rational points.
The purpose of this paper is to study maximal curves, that is, curves X over F q whose number of rational points #X(F q ) reaches the Hasse-Weil upper bound.In this case one knows that q must be a square.
Let k be the finite field with q 2 elements, where q is a power of a prime p.Let X be a projective, connected, non-singular algebraic curve defined over k which is maximal, that is, #X(k) satisfies #X(k) = q 2 + 2gq + 1. (0.1) Let P ∈ X(k) and set D = g n+1 q+1 the k-linear system on X defined by the divisor (q+1)P .Then n ≥ 1, and D is independent of P .In fact D is a simple base-point-free linear system on X (Corollary 1.2.3,Remark 1.2.5 (ii)).This allow us to apply Stöhr-Voloch's approach concerning Weierstrass point theory over finite fields [S-V].Moreover, the dimension n+1 of D and the genus g are related by Castelnuovo's genus bound for curves in projective spaces ( [C], [ACGH,p.116],[Ra,Corollary 2.8]).
The paper was written while the first author was visiting the Instituto de Matemática Pura e Aplicada, Rio de Janeiro (Oct. 1995-Jan. 1996).This visit was supported by Cnpq.
The second author is supported by a grant from the International Atomic Energy Agency and UNESCO.
Moreover by using the already mentioned Castelnuovo's bound one can prove that 4g > (q − 1) 2 if and only if n = 1.Therefore, we assume from now on that n ≥ 2.
The Hermitian curve is a particular case of the following type of curves.Let m be a positive divisor of q + 1, and let consider These curves are maximal ([G-V, Thm.1]) and have very remarkable properties (see e.g [G-V], [Sch]).
Under a hypothesis on non-gaps at rational points we prove that maximal curves are k-isomorphic to H m,q for some m ∈ Z + .Theorem 0.1.Let X be a maximal curve of genus g > 0. Assume that there exists P 0 ∈ X(k) such that the first non-gap m 1 at P 0 satisfies where n + 1 is the dimension of the complete linear system defined by (q + 1)P 0 .Then one of the following possibilities is satisfied From this theorem and a result due to Lewittes (see inequality 1.6) we obtain an analogous of the main result in [F-T]: Corollary 0.2.Let X be a curve satisfying the hypotheses of Theorem 0.1.Let t ≥ 1 be an integer, and suppose that the genus g of X satisfies Then one of the following conditions is satisfied Remark 0.3.In case nm 1 = q the authors actually conjecture that then 2g = (m 1 − 1)q, and X is k-isomorphic to a curve whose plane model is given by F (y) = x q+1 , where F (y) is a F p -linear polynomial of degree m 1 .But we have not yet been able to prove this.We notice that the veracity of this conjecture implies t = n and 2g = ( q t − 1)q = (m 1 − 1)q in the statement (ii) of the above corollary.

Preliminaries
Throughout this paper we use the following notation: • k denotes the finite field with q 2 elements, where q is a power of a prime p. k denotes its algebraic closure.
• By a curve we mean a projective, connected, non-singular algebraic curve defined over k. • The symbol X(k) (resp.k(X)) stands for the set of k-rational points (resp.the field of k-rational functions) of a curve X.
) denotes the divisor (resp.the polar divisor) of x.
• Let P be a point of a curve.v P (resp.H(P ))) stands for the valuation (resp.the Weierstrass semigroup) associated to P .We denote by m i (P ) the ith non-gap at P .• Let D be a divisor on X and P ∈ X.We denote by deg(D) the degree of D, by Supp(D) the support of D and by v P (D) the coefficient of P in D. If D is a k-divisor, we set L(D) := {f ∈ k(X) : div(f ) + D 0}, and ℓ(D) := dim k L(D).The symbol "∼" denotes module linear equivalence.
• The symbol g r d stands for a linear system of dimension r and degree d.
1.1.Weierstrass points.We summarize some results from [S-V].Let X be a curve of genus g, D = g r d a base-point-free k-linear system on X.Then associated to P ∈ X we have the hermitian P -invariants j 0 (P ) = 0 < j 1 (P ) < . . .< j r (P ) ≤ d of D (or simply the (D, P )-orders).This sequence is the same for all but finitely many points.These finitely many points P , where exceptional (D, P )-orders occur, are called the D-Weierstrass points of X. (If D is generated by a canonical divisor, we obtain the usual Weierstrass points of X.) Associated to D there exists a divisor R supporting the D-Weierstrass points.Let ǫ 0 < ǫ 1 < . . .< ǫ r denote the (D, P )-orders for a generic P ∈ X.Then for P ∈ X, for each i, and Associated to D we also have a divisor S whose support contains X(k).Its degree is given by where the ν ′ i s is a subsequence of the ǫ ′ i s.More precisely there exists an integer I with 0 < I ≤ r such that ν i = ǫ i for i < I and ν i = ǫ i+1 otherwise.Moreover for P ∈ X(k) we have 1.2.Maximal curves.Let X be a maximal curve of genus g.In this section we study some arithmetical and geometrical properties of X.To begin with we have the following basic result which is containing in the proof of [R-Sti, Lemma 1].For the sake of completeness we state a proof of it.
Lemma 1.2.1.The Frobenius map Fr J (relative to k) of the Jacobian J of X acts just as multiplication by (−q) on J .
Proof.All the facts concerning Jacobians can be found in [L,VI,§3].Let ℓ = p be a prime and let T ℓ (J ) be the Tate module of J .Then the characteristic polynomial P (Fr J )(t) of the action Fr J on T ℓ (J ) is equal to t 2g L(1/t) where L(t) denotes the numerator of the Zeta function of X.Since X satisfies (0.1), L(t) = 2g i=1 (1 + qt) and thus P (Fr J )(t) = (t + q) 2g .Now we know that Fr J is diagonalizable [Ta,Thm. 2] and all its eingenvalues are −q.This means that Fr J acts just as multiplication by −q on T ℓ (J ).Finally since the natural homomorphism of Z-algebras is injective, the proof follows.
Now fix P 0 ∈ X(k), and consider the map f = f P 0 : X − → J given by P − → [P − P 0 ].We have f • Fr X = Fr J • f, where Fr X denotes the Frobenius morphism of X relative to k. Hence from the above equality and Lemma 1.2.1 we get Corollary 1.2.2.
In fact, set i := min{j ∈ Z + : Fr j X (P ) = P }.Now applying Fr i−1 X * (see [Har,IV,Ex. 2.6]) to the equivalence in Corollary 1.2.2 we get Fr X (P ) + (q − 1)P ∼ qFr i−1 X (P ).Now the remarks follows from the fact that Fr i−1 X (P ) = Fr X (P ) if and only if P ∈ X(F q 4 ).
Proof.Let P ∈ X.Since j 1 (P ) = 1 (cf.Theorem 1.2.4 (iii)) we know already that π(X) is non-singular at all the branches centered at P .Thus π is an embedding if and only if π is injective.
In both cases we have π −1 (π(P )) = {P }.This means altogether that π is injective and so indeed a closed embedding.
Proposition 1.2.7.Suppose that π : X → P n+1 is a closed embedding.Assume furthermore that there exist r, s ∈ H(P 0 ) such that all non-gaps at P 0 less than or equal to q + 1 are generated by r and s.Then H(P 0 ) is generated by r and s.In particular the genus of X is equal to (r − 1)(s − 1)/2.
2. Proofs of Theorem 0.1 and Corollary 0.2 Set m := m 1 .Recall that n + 1 is by definition the dimension of D := |(q + 1)P | for any P ∈ X(k).Let π be the morphism associated to D. By Remark 1.2.5 (ii) we have nm ≥ q, and hence by the hypothesis on m we get nm ∈ {q, q + 1}.
Moreover, the set of the D-Weierstrass points of X coincides with the set of k-rational points.
Let P ∈ X \ X(k).By Theorem 1.2.4 we have that j n+1 (P ) = q.If v P (y) > 1, then from (2.2) we get nv P (x) = q = mn − 1 and hence n = 1.Since by hypothesis n > 1 then v P (y) = 1.This finish the proof of the lemma.
Let T 1 (resp.T 2 ) denote the number of points P ∈ X(k) whose (D, P )-orders are of type (i) (resp.type (ii)) in Lemma 2.1.2.Thus by (1.2) we have and by Riemann-Hurwitz applied to y : Consequently, since T 1 + T 2 = #X(k) = q 2 + 2gq + 1, from the above two equations we obtain Proposition 2.1.1.Now we are going to prove the uniqueness part of the result.To begin with we generalize [R-Sti, Lemma 5].
Proof.Consider y : X − → P 1 ( k) as a map of degree m = m 1 .From the proof of Lemma 2.1.2we see that y has (q + 1) ramified points.Moreover, all of them are rational and totally ramified.
Let n i = #y −1 (P i ) ≤ m.Since 2g = (q − 1)(m − 1) by Proposition 2.1.1,then we have from where it follows that r = q 2 − q and n i = m for each i.Now it follows that k(X) | k(y) is Galois as in the proof of [R- Sti,Lemma 5].It is cyclic because there exists rational points that are totally ramified for y.Proposition 2.1.4.Let X be a curve as in Proposition 2.1.1.Then X is k isomorphic to H m 1 ,q .
Proof.Let y be as in (2.1).
Claim 1. X has a model plane given by an equation of type where f ∈ k[T ] with deg(f ) = q, f (0) = 0, and v ∈ L(qP 0 ).
Proof.(Claim 1.) We know that k(X) | k(y) is cyclic (Lemma 2.1.3).Let σ be a generator of k(X) | k(y).Set V := L(qP 0 ), U := L((n − 1)mP 0 ).Then σ | V ∈ Aut(V ) and σ | U = id | U. Since p ∤ m we then have that σ | V is diagonalizable with an eigenvalue λ a primitive m-root of unity in k.Let v ∈ V \ U be the corresponding eigenvector for λ.Now since Norm k(X)|k(y) (v) = −v m and since v ∈ L(qP 0 ) we conclude the existence of f ∈ k[T ] such that f (y) = v m and deg(f ) = q.Finally from the fact that y has exactly (q + 1) rational points as totally ramified points, it follows that f splits into linear factors in k[T ].Hence we can assume f (0) = 0. Now from the claim in the proof of Lemma 2.1.3,Claim 1 and nm = q + 1 it follows that f n (α) − f nq (α) = 0 for α ∈ k, and hence we obtain f n (T ) ≡ f nq (T ) (mod T q 2 − T ). ( * ) Set f (T ) = q i=1 a i T i , f n (T ) = nq i=1 b i T i .Claim 2. a 1 = 0, a i = 0 for 2 ≤ i ≤ q − 1.
If one can show that π : X → P n+1 is a closed embedding, then from Proposition 1.2.7 we would have 2g = q(m 1 (P ) − 1) for P ∈ X(k).
Remark 2.2.1.The hypothesis on the first non-gap of Theorem 0.1 is necessary.In fact, consider the curve from Serre's list (see [Se,§4]) over F 25 , g = 3.Then it is maximal.Let m, 5, 6 be the first non-gaps at P ∈ X(F 25 ).If m = 3 from Theorem 0.1 (i) we would have g = 4. Thus m = 4.